Optimal Trading Strategies
in a Limit Order Market with Imperfect Liquidity
P. Kovalevaa,∗, G. Iorib
a City
b City
University, Department of Economics, D110 Social Sciences Bldg, Whiskin St., London EC1R 0JD
University, Department of Economics, D304 Social Sciences Bldg, Whiskin St., London EC1R 0JD
Abstract
We study the optimal execution strategy of selling a security. In a continuous time diffusion framework, a risk-averse
trader faces the choice of selling the security promptly or placing a limit order and hence delaying the transaction in
order to sell at a more favourable price. We introduce a random delay parameter, which defers limit order execution
and characterizes market liquidity. The distribution of expected time-to-fill of limit orders conforms to the empirically
observed exponential distribution of trading times, and its variance decreases with liquidity. We obtain a closed-form
solution and demonstrate that the presence of the lag factor linearizes the impact of other market parameters on the
optimal limit price. Finally, two more stylised facts are rationalised in our model: the equilibrium bid-ask spread
decreases with liquidity, but increases with agents risk aversion.
JEL classification: D4, D81, D84, G1, G12
Keywords: order submission, execution delay, first passage time, risk aversion, liquidity traders
1. Introduction
The problem of optimal order placement is the kernel of the successful implementation of an investment strategy
since the optimal trade execution reduces the associated transaction costs and augments expected returns. Traders
construct their submission strategies to benefit from particular market properties and order types hence the architecture
of the market defines their expectations about the future price dynamics and trade execution efficiency. In pure
quote-driven dealer markets small orders typically execute at the best opposing dealer quote regardless of the order
type. In public limit order books, market orders may encounter price improvement, whereas limit orders execution
is conditional upon where traders place their limit prices relative to the prevailing bid and ask. The goals pursued
by traders operating in a competitive environment of non-intermediated limit order markets are distinct from those of
market makers. The fundamental distinction between the models of dealer quoting and that of a generic trader problem
is that the former is essentially indifferent to execution, therefore his objective function is a zero profit. In limit order
markets agents have multiple reasons to trade and various financial instruments available to execute a particular trade.
Primarily, however, each trader entering a non-intermediated market must decide upon the type of order to use – a
market order or a limit order.
Substantial empirical evidence suggests limit orders play a dominant role in the markets. For instance, Harris and
Hasbrouck (1996) found that limit orders generally perform best even in the presence of a nonexecution penalty and
market price improvement. According to Guéant et al. (2011), more than half of all trades, approximately 60%, are
as passive as possible, that is they fill the queue to trade rather than consume the liquidity. Biais et al. (1995) present
an empirical analysis of the order flow of the Paris Bourse which is a pure limit-order market. They find that traders’
strategies vary with market conditions, with more limit orders at times when spreads are wide and more market orders
author. Tel: + 44 (0)20 7040 4645, Fax: + 44 (0)20 7040 8580.
Email addresses: [email protected] (P. Kovaleva), [email protected] (G. Iori)
∗ Corresponding
Preprint submitted to Elsevier
May 15, 2012
at times when spreads are narrow. In real markets the motivation to choose a limit order over a market order is
obfuscated by various subtle effects, for example: the discrepancy in transaction fees,1 the option to submit multiple
orders simultaneously, the existence of several distinct markets for the same asset and the possibility to withdraw
the order. Many previous studies were based on the division of the traders into two main groups: liquidity suppliers
who trade with limit orders, and liquidity demanders who have higher immediacy priority. In contrast, we develop
a framework where depending on the present market conditions the trader sets out either to provide or to consume
liquidity.
In practice, even though investment and trading decisions are formulated simultaneously, they are usually analyzed
and accomplished somewhat in isolation. Our model reflects precisely such allocation of tasks since we do not take
into account the portfolio the trader might hold and how the outcome of the trading operations affects the balance and
the value of this portfolio. Instead, we assume that the agent enters the market with a given trading goal and his task
is to devise an optimal execution strategy given market characteristics at the time of his arrival. An important issue
in market microstructure is the information set available to traders. There is abundant empirical evidence suggesting
that market movements are often triggered by information updates and the presence of information asymmetries. A
number of early studies showed that information component of the bid-ask spread is fractional; according to Huang
and Stoll (1997), on average it compounds less than 12% of the spread. Yet, more contemporary findings indicate
that the information component of the spread is significant and amounts for as much as 80% (Gould et al., 2010).
Information asymmetry is usually defined in a fairly broad sense and it does not necessarily imply, for instance, any
form of legal or illegal insider trading. Simply because the tools with which market participants assess the market
vary, they draw different predictions from the same market conditions. However, superior or rather heterogeneous
information is not the only basis for trading as was demonstrated by Milgrom and Stockey (1982). Guided by this line
of thought, we do not analyze information effects per se; we incorporate traders’ individual expectations of the future
asset price in our model, while leaving the grounds for this valuation beyond the scope of the present study.
In the present study we introduce a tractable parameter to characterize limit orders. More precise, we consider
a problem of a trader who has to liquidate a position in an actively traded asset within a given period of time and
forms his strategy based upon the market dynamics represented by bid and ask prices. Building upon the model
in Iori et al. (2003), we conceptually improve it by incorporating an exogenous limit order execution factor – an
exponential random delay. Moreover, we use a quadratic utility function and examine how different risk perception
affects the optimal strategy of a trader in a limit order market and his average waiting time. This formulation proves
more advantageous both in terms of interpretation and the ease of potential calibration to data. Further we provide an
explicit static solution to the limit order trading problem for the quadratic utility preferences and subsequently identify
the key determinants of the limit order attractiveness to the trader.
The remainder of this paper proceeds as follows. Section 2 reviews relevant theoretical and empirical literature.
Section 3 describes the design of a market in which traders submit their orders and outlines the clearing mechanism.
Section 4 presents the problem of a risk-averse trader who operates in this market and the expected payoffs associated
with particular execution mechanisms. Section 5 reports comparative statics analysis for the parameters of the model.
The subsequent Sections 6 and 7 analyse two special cases and the distribution of the waiting time respectively. The
existence of equilibrium spread and the appropriate conditions are discussed in Section 8. Section 9 concludes.
2. Literature Review
There is an extensive literature on the subject of the optimal order submission strategy in limit order markets. The
main distinction between the theoretical approaches adopted in various studies lies in the definition of the limit order
execution mechanism and, consequently, the resulting probability distribution.
Equilibrium analysis of order-driven markets has been realized by Kumar and Seppi (1994); Chakravarty and
Holden (1995); Parlour (1998); Foucault (1999); Foucault et al. (2005); Goettler et al. (2005) and Rosu (2009). All
of these models are variants of a dynamic multi-agent sequential bargaining game where heterogeneous traders derive
their best-response order submission strategies. Parlour (1998) assumes that the probability of execution of a sell
1 Maker/taker market microstructure, for instance, encourages liquidity provision by offering a fee discount when a limit order is submitted,
while consuming liquidity entails higher charges.
2
limit order depends on the arrival of buy market orders and the relative attractiveness of buy market orders depends
on relative attractiveness of buy limit orders, thus execution probabilities of buy and sell limit orders are determined
jointly over time. In situations when prices are fixed, and that should be the case in equilibrium, optimal order
placement is contingent upon a single factor – the distribution of agents’ impatience characteristic. Allowing for price
movements, Foucault (1999) describes the asset price via binomial model and assumes that limit orders are valid only
for one period, therefore, at each point in time the book is either full or empty. In this setting the probability of limit
order execution is endogenous and part of execution risk arises from the next trader’s order type. The focus on the
optimal behavior in equilibrium yields numerous implications coherent with the documented market observations and
these models proved especially useful for policy-makers. However, order-driven markets do not seem ever to attain
these conditions: while in equilibrium all market participants should get zero profits, the depth of real limit order book
is usually insufficient to drive average expected profits to zero.
A separate branch of optimal order placement literature, where the individual traders’ order submissions are
aggregated by the asset price dynamics, was initiated by Cohen et al. (1981). In their seminal paper, Cohen et al. (1981)
propel the theoretical analysis of optimal choice between market and limit orders in a framework with the probability
of order execution contingent upon future price movements and the associated probability densities. Trading takes
place when the trajectory of the best quote first crosses the barrier determined by the limit price. Cohen et al. (1981)
model the security price with a compound Poisson process, which, by the very definition, invokes a jump in the
probability of execution: if a time-constrained trader is willing to buy a stock via limit order and sets a price infinitely
close to the current best ask, the probability of trading never attains unity. This property permitted to establish a
so-called “gravitational pull effect”: when the bid-ask spread is narrow, the benefit of a price improvement with a
limit order becomes small compared to the risk of non-execution so traders are pulled to use market orders instead.
Consequently, Cohen et al. (1981) argue that limit order strategy is not always superior to trading with market orders
or not submitting any orders at all. This model adequately captures the trade-off between favorable price and order
execution probability and, in this sense, draws a line between market and limit orders.2 Cohen et al. (1981) further
demonstrate that as the order arrival rate increases the Poisson process converges to the Wiener process eliminating the
discontinuity in the execution probability function. The model setup, however, is too complex to obtain a closed-form
solution and their analysis remains qualitative for the main part.
Langnau and Punchev (2011) concentrate deliberately on the issue of adequate price modeling in a public limit
order book. Adopting the results achieved by Kou and Wang (2003), they compare the implications of a pure diffusion
and a double exponential jump diffusion mid-point price generating processes. The most appealing properties of the
double exponential jump distribution include the memorylessness of the price and the ability to accommodate the leptokurtic nature of returns. They arrive at the closed-form solution for the first passage time with distinct expressions
for buy and sell strategies in the latter case. The compelling result of this paper is that the jump diffusion specification
accommodates the asymmetric shape of a limit order book as well as a fat-tailed distribution of log returns. Therefore, Langnau and Punchev (2011) establish that exponential double jump diffusion specification provides a more
accurate description of the markets and is compatible with the equilibrium conditions whereas log-normal prices are
not. The jump diffusion setting requires to calibrate or fit considerably more parameters thus impeding its practical
implementation.
Given the analytical complexity of the setting with the jumps in prices it seems reasonable to address the problem
of optimal strategy within the context of a continuous time diffusion price. Several contributions can be found in the
financial and econometric literature. For instance, Iori et al. (2003) show that despite its obvious shortcomings, the
log-normal price still suggests that optimal limit order strategy is coherent with the traders behavior observed from the
market. A mean-reverting price specification, particularly relevant for commodities, depicts an interesting cross-over
effect: the optimal value of the strategy increases with the speed of reversion for small expiry times, while decreasing
for big expiry times. Nevertheless, pure first passage time models such as the latter cannot justify the existence of
spread due to the inability to differentiate between a marketable limit order and a market order, or more precisely, to
distinguish among the time-to-first-fill, time-to-completion, and time-to-censoring when a limit order is withdrawn.
An attempt to preserve the appealing mathematical lightness of first passage time models and bridge it with the
notion of imperfect liquidity was made by Harris (1998). Harris (1998) improves the framework with a pure diffusion
2 The
orders placed inside the current spread are also interpreted as limit orders.
3
prices by adding a supplemental criterion – an aggregated factor of degree of execution difficulty. The degree of
difficulty in limit order execution is defined as an additional barrier which a limit price has to pass before a limit
order is filled. Therefore it is not sufficient to become the best price on the same side of the market, a limit order
has to supersede this best price: for instance, a sell limit order is executed when submitted limit order price is lower
than best ask less the difficulty parameter. Two execution mechanisms were juxtaposed: with certainty and with some
probability. Comparative statics analysis based on a numerical solution confirms that the probability of order execution
depends positively on this difficulty factor, in the certainty case as well. An alternative stylized interpretation of the
problem was given by Hasbrouck (2006). Assuming that traders on the opposite side of the book are ascribed with
unobserved reservation prices, a random collateral barrier following exponential distribution is analyzed. Hasbrouck
(2006) argues that the number of potential counterparties is decreasing in the intensity rate of reservation prices
distribution, thus optimal strategy becomes more aggressive. Although very valid, theoretical approaches presented
by Harris (1998) and Hasbrouck (2006) have certain limitations. It appears to be a challenging task to infer the
reservation prices or the degree of difficulty in limit order distribution from data due to the apparent difficulty in
disentangling the impact of these external factors from the order aggressiveness determinants and the actual trading
times.
Empirical research also reveals the bilateral causality in the relation between execution time and limit order submissions. Using survival analysis Lo et al. (2002) demonstrate that limit order execution times are increasing as the
limit prices become more aggressive. This is both an outcome of the price priority rule and the potentially endogenous
effect of aggressive order creating higher demand for trade from the other side of the book, as suggested by Parlour
and Seppi (2008). Moreover, Lo et al. (2002) estimate the cumulative probability densities of the actual limit order
execution times and compare to their hypothetical counterpart calculated as the first hitting times of geometric Brownian motion. The latter appears to understate largely expected trading times. Histograms of time-to-execution for
limit orders exhibit exponential distribution and a comparison reveals that they differ not only in one or two moments
but over their entire support. Tkatch and Kandel (2006) apply a simultaneous equations methodology to the Tel Aviv
Stock Exchange data and find a significant causal impact of expected execution time on investors’ decisions of which
orders are submitted. Without analysing directly how a particular trader’s order placement affects the market, our
model captures the statistical properties of trading times and relates the expected time-to-fill to order aggressiveness.
3. The Model
3.1. The Market
We consider the problem of an investor who has a position in a traded asset which has to be liquidated within
a pre-specified time horizon. One option the agent has is to use a market order and trade at the best available price
at once. Alternatively, he can submit a limit order and hope to trade at a more favorable price. In a double auction
market a transaction occurs when a market order hits the quote on the opposite side of the book. We assume that there
is no information asymmetry and future price dynamics depend only upon public information.
Market orders trade at the best price currently available at the market and is filled instantaneously. The actual
execution price of a market order is subject to the current market situation – this aspect of trading is referred to as
execution price uncertainty. On the one hand, if the trade is negotiable then the agent can obtain a price improvement:
in a situation when the spread is wide a trader may gain the speed in order execution by stepping inside the spread and
so compromising on the price. On the other hand, market order traders bear the risk of trading at worse prices than
they initially expect due to the rapidly changing market environment.
Limit orders are instructions to trade at the best price available but only if it is not worse than the limit price
specified by a trader. The probability that a particular order will be filled depends on its limit price. Marketable limit
orders, being the most aggressive, are easiest to fill. There exist two main risks associated with limit order trading:
execution uncertainty when market moves away from the submitted limit price hence the agent never trades and ex
post regret when for various reasons prices move towards and through the limit price.
Without loss of generality we further focus on the problem of a limit sell order, therefore optimizing the limit ask
price Ka . The position in a security must be liquidated within a fixed period of time T . Its current price is determined
by the best bid b0 , the highest buying price, and the best ask a0 , the lowest selling price. A pair of stochastic log-normal
4
processes bt and at describe the trajectories of the bid and ask prices in the book respectively:
dat = µa at dt + σa at dWta
(1)
dbt = µb bt dt + σb bt dWtb
(2)
h
i
where Wta and Wtb are ρ-correlated standard Brownian motions, E dWta dWtb = ρdt.
We analyze short-term decisions when market conditions do not change substantially, so the assumption deterministic price trends is viable. Rosu (2009) observed through a theoretical dynamic model, that there is a co-movement
effect between bid and ask: when the bid lowers after a sell market order arrives, the ask also decreases but by a
smaller amount, therefore widening the spread. Therefore, we should expect to observe a positive correlation between
the best bid and ask price movements in real markets.
3.2. Order Execution Criteria
When a trader implements a limit order and submits his preferred ask price Ka to the sell side of the market,
he is “competing” with other potential sellers, or the best ask process at . Placing a limit order far from the current
quotes implies more severe competition and increases the chances that the opportunity to trade will not arise before
expiry time. Intuition suggests that when the market is illiquid and there are many gaps in the book, moving away
from current prices is less risky. Also, the probability of execution of more aggressive limit orders increases, since
normally the gaps get filled if they approach the best price. Once all the offers to sell at a lower price are picked up
by arriving buyers, then his limit sell price becomes the best ask in the market. This is a general operation principle
of double auction markets, however, the subsequent order matching and clearing is a less trivial process. Apart from
competing in price, traders also contend in order size and time priority. Consequently, for modeling purposes, we
need to delineate some mechanical order execution rule which will take into account such aspects of trading.
The market we study is sufficiently liquid, implying that the price dynamics can be described by a continuous
process. Limit orders in the book are executed in first-in, first-out time priority. Once the limit order of an agent
becomes the best price in the market, a random delay before trading ensues. The delay is usually small relative to the
time horizon, but there is a small probability that the delay will be sufficiently long and hence distort the schedule of
the agent’s trading operations. The transaction occurs with an unforeseen delay ε which is independent of the asset
prices and is modeled as an exponential random variable with constant intensity λ. The delay in trading occurs due to
the fact that somebody else might have put an order at exactly the same price but earlier than the agent in question.
Another argument for delay, albeit a minor one in the context of modern electronic markets, is an operational delay.
Clearly, in most of the cases, once this market is trading close enough to the quote that an agent has previously
submitted, it will not move away swiftly. More importantly, this market design implies that the existence of a spread
is related to the costs of waiting: even if the trader’s horizon is long enough the delay might force him to use a market
order and pay the spread despite initially high chances of trading at a more favourable limit price .
In their empirical study based on survival analysis Lo et al. (2002) observe exponentially distributed trading times.
The absence of the peak near zero can by partially attributed to the discrete nature of the data, whereas our result
applies to continuous time. Cho and Nelling (2000) argue that given market orders arrive in a non-homogeneous
Poisson process the waiting times of a limit order follow a Weibull distribution. The estimations from a duration
model (based on TORQ data from Harris and Hasbrouck (1996)) suggest that a histogram of empirical observations
resembling an exponential probability density as well as Weibull. Notably, our theoretical result is compatible with a
Weibull distribution specification for a shape parameter smaller or equal to unity. Cincotti et al. (2005) use tick data
from 7 different US financial markets, they find that the distribution of trade waiting times is a mixture of exponential
processes and verify this implication on an agent-based artificial market model.
4. Agent’s Problem
The trader is alloted the task of selling one unit of the asset and has to complete the trade by a certain deadline. We
assume that in order to optimize the price he is willing to adopt a limit order strategy but is aware that there is a penalty
for the non-execution event. Therefore, we introduce the possibility of converting to a market order at maturity if his
5
limit order was not filled. If his limit order does not reach the front of the queue in the limit order book within the
horizon T , the agent has to sell the asset trading at the best available bid.
In a perfectly liquid market even if the initial limit order at a certain price Ka > a0 was not picked before T ,
submitting a sell limit order at the best ask guarantees trading immediately. Thus, a time-zero discounted payoff of
the agent’s strategy equals the liquidation value of the asset:
V (Ka , a0 ; T ) = e−δτ Ka I{τ≤T } + e−δT aT I{τ>T } ,
(3)
where τ = inf{t ≥ 0 : at = Ka }. Essentially, in the perfectly liquid market, at = bt must hold at all times because the
execution mechanisms is exactly the same at both prices.
In a situation with a random delay the agent cannot bear additional risk at maturity. Hence submitting a limit
order is not a sure trade and he is obliged to use market order instead and sell the security at the highest price on
the buy side of the market at time T – that is the best bid bT . Suppose a trader decides to place a limit order inside
the spread, then his only concern is a random delay matters.3 However, placing an order at the best ask entails same
expected time-to-fill but at a more beneficial price. Therefore, in this stylised framework no trader will be willing
to sell security even one tick cheaper than the current best ask4 This strategy yields the following expression for the
discounted terminal payoff:
Vλ (Ka , a0 , b0 ; λ, T ) = e−δ(τ+ε) Ka I{τ+ε≤T } + e−δT bT I{τ+ε>T } ,
(4)
with τ = inf{t ≥ 0 : at = Ka } and ε ∼ Exp (λ).
Numerous studies solve for the optimal strategy from the stand-point of a risk-neutral agent. However, if risks
cannot be hedged away, the trader is concerned not only about the expected payoff but also about the range where the
future payoff might lay, therefore he should take into account the variance in the future wealth. We assume that the
trader is ascribed with a ϕ degree of risk-aversion and maximizes a mean-variance utility function:
EU (Ka ) ≡ max {E [V] − ϕ · Var [V]} .
Ka ≥a0
(5)
Proposition 1.
Denote the power function of the profit from selling a unit of security as Gλ (γ) ≡ (V (Ka , a0 , b0 ; λ, T ))γ .
Given that market prices are positively correlated log-normal processes (1) and (2), the expected value of G for selling
at a limit price Ka ≥ a0 in a market with a random delay in limit order execution is given by expression:5

!A +A
!
!A −A
!
−x − A2 σ2a T
−x + A2 σ2a T
Ka 1 2
λ  Ka 1 2
γ
N
+
N
(6)
EGλ (γ) = Ka
√
√

λ + γδ a0
a0
σa T
σa T

!
!
!
!A −A
 Ka A1 +A3
−x − A3 σ2a T
−x + A3 σ2a T 
Ka 1 3
−(λ+γδ)T 


− e
+
N
N
√
√

a0
a0
σa T
σa T

!
!2A
!
 x − A4 σ2a T
−x − A4 σ2a T
Ka 4
γ γ(µb +(γ−1)σ2b /2−δ)T 
N
+ b0 e
−
N
√
√
a0
σa T
σa T

!
!
!A −A
!
 Ka A4 +A5
−x − A5 σ2a T
−x + A5 σ2a T 
Ka 4 5

N
+
N
+ e−λT 
√
√
a0
a0
σa T
σa T
where x = ln Ka0a and the constants are calculated as follows:
√
√
(µa −σ2a /2)2 +2γδσ2a
(µa −σ2a /2)2 −2λσ2a
µ −σ2 /2
A1 = a σ2a , A2 =
,
A
=
,
3
2
σ
σ2a
a
a
√
2
2
2
2
(µa −σa /2+γρσa σb ) −2λσa
µ −σ /2+γρσ σ
A4 = a a σ2 a b , A5 =
;
σ2
a
a
3 Mike and Farmer (2008) even argue that in determining a trade-off between the price improvement and execution risk, the distributions of limit
prices in the book and inside the bid-ask spread differ qualitatively. The reason is that the first-in-first-out rule does not apply in the latter case.
4 We actually impose in this way that time increments are not equally spaced but the intervals between the price moves.
5 The mean and the variance of the expected profit from selling the asset are: EV = EG (1) and Var[V ] = EG (2) − (EG (1))2 .
λ
λ
λ
λ
λ
6
(µ −σ2 /2)2
and ρ ≥ 0.
λ ∈ 0; a 2σa2
with the parameter constraints
a
Proof. See Appendix A.1.
Consider the case when alimit order to
sell is submitted exactly at the current ask. The expected value of this
λ
−(λ+δ)T
strategy is EVλ |Ka =a0 = a0 λ+δ 1 − e
+ b0 e(µb −δ−λ)T . The greater is the average delay E[ε] = 1/λ, the higher is
the weight given to the profit from using a market order at maturity. If λ = 0, that is, the delay is substantially long,
then the expected payoff is exactly the discounted expected bid EV0 |Ka =a0 = e−δT E [bT ].
Regarding the upper bound of delay parameter, an intuitive interpretation for this benchmark lies in its relation to
the risk-adjusted mean squared to the variance of τ, which is referred to the squared Sharpe ratio. This upper bound
is rather low for realistic values of the parameters and implies a strong liquidity restriction, however, qualitatively the
results appear to be very similar to those based on formula (7).
Proposition 2. Denote the power-law function of the profit from selling a unit of security as Gλ (γ) ≡ (V (Ka , a0 , b0 ; λ, T ))γ .
Given the market prices are ρ-correlated log-normal processes (1) and (2), the expected value of function G for selling
at a limit price Ka ≥ a0 in a market with a small random delay in limit order execution converges to the value:

!A1 +A2
!
!A −A
!
−x − A2 σ2a T
−x + A2 σ2a T 
Ka 1 2
 Ka
γ
+
(7)
N
N
lim EGλ (γ) = Ka 
√
√

λ→∞
a0
a0
σa T
σa T

!
!2A
!
 x − A4 σ2a T
−x − A4 σ2a T 
Ka 4
γ γ(µb +(γ−1)σ2b /2−δ)T 


−
N
+ b0 e
N
√
√
a0
σa T
σa T
√
K (µa −σa2 /2)2 +2γδσa2
µa −σ2a /2
µ −σ2 /2+γρσ σ
a
where x = ln a0 and A1 = σ2 , A2 =
, A4 = a a σ2 a b .
σ2
a
a
a
Proof. See Appendix A.2.
Notably, this result replicates the valuation formula derived in Iori et al. (2003) for the market where limit orders
trade upon achieving the beginning of the queue on the relevant side. However, in accordance with the payoff in
equation (3), in an infinitely liquid market a trader will resolve to a limit order at aT in the non-execution event rather
than pick the best buying order at bT . Nonetheless, as we show later, this setting actually requires a zero bid-ask
spread, or at = bt , ∀t.
In order to assess approximately the expected utility we compute its upper and lower bounds.6 Fig. 1 captures
the convergence of the actual expected utility value and the boundary conditions as distance-to-fill of a limit order
increases. While for a risk-neutral traders the upper and lower bounds appear to be symmetric (Fig. 1a), this is not the
case for positive ϕ where the lower bound becomes significantly nonlinear and more restrictive (Fig. 1b). The reason
for the latter is that for more aggressive orders (when Ka /a0 is small) the probability of limit order execution is high
and expected time to reach the beginning of the queue is relatively short, thus the approximation of penalty term (B.6)
yields a relatively small value and (B.8) – a relatively large.
5. Comparative Statics
5.1. Baseline Parameters
Using analytical expressions (7) and (6) the expected payoff in a perfectly and imperfectly liquid markets respectively, we find the optimal limit price which a trader should submit to achieve the maximum level of expected
mean-variance utility. Further, we implement numerical integration with recursive adaptive Simpson quadrature rule
to obtain the solution to the optimization problem for medium levels of liquidity (λ̄ < λ ∞).7 We then examine the
impact of various model parameters on the optimal decision of a trader.
6 The
derivations are given in Appendix B.2.

2 2 
2
2


7 λ̄ = min  µa −σa /2 , µa −σa /2+γρσa σb 


 2σ2a

2σ2a
7
(b)
(a)
25
2
upper bound
true value
lower bound
upper bound
true value
lower bound
20
1.5
EUλ /a0
EU/a0
15
1
10
5
0.5
0
0
1
1.05
1.1
−5
1
1.15
Ka /a0
1.05
1.1
1.15
Ka /a0
Figure 1: The expected utility of a sell limit order strategy for as a function of the limit price: (a) for a risk-neutral
agent (ϕ = 0), (b) for a risk averse trader (ϕ = 0.01). Results are shown for a0 = 1000, b0 = 995, ∆ = 1,
T = 5 days, µ = 10%, σ = 20%, ρ = 0.1, λ → ∞.
For the purposes of compatibility and comparison with the previous work we set the time-zero best ask in the
book equal to a0 = 1000 with the initial spread of 5. The tick size appropriate for this price range is ∆ = 18 , and
we choose a time horizon T = 5 days in a benchmark case. The discount factor in this market is δ = 5% per annum
while the expected drift parameters are µ = µa = µb = 10% and the annual volatility of σ = σa = σb = 20% with a
small positive correlation in prices ρ = 0.1. It follows immediately from the assumption of no information asymmetry
that all traders are equally informed about the value of the security. The bid and ask prices represent valuation on
demand and supply sides of the same asset and should not diverge significantly at any point in time. Given these
considerations, we work with the cases when the drifts of bid and ask are the same in order to preserve stationarity of
the spread.9 The delays are characterized by the liquidity rate λ = 8 per day10 and risk-aversion parameter is ϕ = 0.01
if not specified otherwise.
5.2. Comparative Static Effects
Prior to the discussion of the effects of various market parameters on the optimal order placement we look at the
shape of the expected quadratic utility function. There are three possible market patterns: sideway market, up-trend
market and down-trend market. A straightforward interpretation of the drift is to link it to the extent of traders’
8 For instance, according to BATS tables the common tick size for FTSE 100 and FTSE 250 excluding high liquidity segment stocks trading in
a price range between £1,000 and £4,999 is £1.
h i
9 Condition µ = µ implies that a process b /a is a martingale and E bt = b0 .
a
t t
b
at
a0
10 Given that a random delay follows an exponential distribution, its mean value is the inverse of λ, so a small expected delay coincides with large
values of λ. The trading span in most markets is 8 hours, so we assume that λ = 32 corresponds to a 15-minutes mean delay, λ = 96 – to 5-minutes,
λ = 480 – to 1-minute, etc.
8
optimism or pessimism about the future price movements based on the past market performance. The greater the
expected surge in the asset price, the higher the chances of a sell limit order getting filled before expiry. Our analysis
concerns trading in a risky asset, so a drift higher than the discount factor δ = 5% implies a rise in stock price. This,
in turn, makes limit orders more attractive for the investor who is willing to sell.
(b)
(a)
1.01
µ = 0%
µ = 5%
µ = 10%
1.0015
1.005
EUλ /a0
1.001
EU/a0
µ = 0%
µ = 5%
µ = 10%
1.0005
1
0.9995
1
0.995
0.999
1
1.05
1.1
0.99
1
1.15
1.05
1.1
1.15
Ka /a0
Ka /a0
Figure 2: The expected utility of a sell limit order strategy for a risk-neutral trader (ϕ = 0) as a function of the limit
price: (a) no delay in execution, (b) in presence of a random delay in limit order execution. Results are shown for
a0 = 1000, b0 = 995, ∆ = 1, T = 5 days, σ = 20%, ρ = 0.1, λ = 8.
We compare the expected utilities of a risk-neutral (Fig. 2) and risk-averse (Fig. 3) traders. The expected profit
that a risk-neutral agent attains trading in a perfectly liquid limit order book is the same for any limit price he might
choose to submit (Fig. 2a) if the expected return of the underlying security equals risk-free rate. If the price trend is
downward sloping – the best choice would be to sell at the current prevailing price, while upward trend implies that a
trader should use an infinitely high price. This relationship matches perfectly the result obtained by Iori et al. (2003).
In contrast, the expected utility of a trader selling via limit orders in an illiquid market retains a concavo-convex shape
for a range of risk aversion coefficients, including ϕ = 0 (Fig. 2b and Fig. 3b). Arguably, placing a limit order close
to the current quote does not eliminate entirely a non-execution risk because of a random delay. On the one hand,
moving away from current ask reduces the probability that the market will reach that level. On the other hand, given
that correlation ρ is small, the price of the market order at maturity is likely to be lower, therefore, the penalty is less
tangible. Also, as Harris and Hasbrouck (1996) confirm, the opportunity cost of unfilled limit orders behind the best
quotes is significant.
Focusing on the effect of the drift parameter we notice that the optimal Ka∗ in a situation without delay and no risk
aversion is much more sensitive to the change in the price drift µ. In effect, the conclusion to be drawn from Fig. 2a
and Fig. 3a is that perfect liquidity implies binary choice: either to trade at the current quote or post an infinitely
high limit sell price. The threshold µ̂ after which an agent switches from a marketable order to a more passive one is
predictably moving rightward as risk aversion raises: it is below zero for ϕ ≤ 0.02 (Fig. 4). Thereupon, in a market
9
(a)
(b)
1.005
1.005
µ = 0%
µ = 5%
µ = 10%
1
EUλ /a0
EU/a0
1
0.995
0.99
0.985
1
µ = 0%
µ = 5%
µ = 10%
0.995
0.99
1.05
1.1
0.985
1
1.15
Ka /a0
1.05
1.1
1.15
Ka /a0
Figure 3: The expected utility of a sell limit order strategy for a risk-averse trader (ϕ = 0.01) as a function of the
limit price: (a) no delay in execution, (b) in presence of a random delay in limit order execution. Results are shown
for a0 = 1000, b0 = 995, ∆ = 1, T = 5 days, σ = 20%, ρ = 0.1, λ = 8.
with imperfect liquidity a trader tends to choose a more passive strategy for certain values of the price drift, while
the optimal decision of the same trader in an absolutely liquid market under the same circumstances is not uniquely
defined. Once a random delay is introduced to the market, the optimal limit price Ka∗ increases linearly with µ, as
Fig. 4 clearly reveals, while perfect liquidity assumes limit prices Ka∗ = {a0 or ∞}. A possible explanation is that an
investor operating in a liquid double auction will take on an opportunity with a limit order as soon as there appears
a slight chance to make higher profit µ > δ since the penalty which non-trade event implies is somewhat artificial –
the worst case scenario is to submit a limit order at maturity at aT (Iori et al., 2003). The situation with delay offers a
more compelling description of trading strategy (Fig. 4), suggesting a gradual increase in limit price with the increase
in price trend.
The limit price is diminishing as the bid-ask spread at the time of order placement increases Fig. 5. Using Paris
Bourse order flow data, Biais et al. (1995) find that limit orders prevail at times of wide spreads and market orders –
at times of narrow spreads. This tendency is compatible with the trading pattern of arbitrageurs and high frequency
traders. However, the validity of our result is in line with Ranaldo (2004) who observes that order aggressiveness of
patient traders increases as spread widens.
Fig. 6 demonstrates the effect of delay intensity on Ka∗ . The more risk-averse the trader, the smaller is an absolute
impact of delay characteristic on the optimum. For relatively high levels of risk tolerance (ϕ = {0; 0.01; 0.02}) there
is a positive relationship between liquidity λ and the optimal limit price Ka∗ . This is an intuitively appealing result: an
increase in intensity rate λ implies a decrease in mean delay (as well as its variance), thus, other things being equal,
leads to a shorter time-to-completion of a trade. The results for the different levels of risk tolerance suggest that an
increase in ϕ decreases the optimal Ka∗ , as expected.
Further, we examine the effect of the correlation between the paths of the best bid and ask prices. In a perfectly
10
1.025
ϕ=0
ϕ = 0.01
ϕ = 0.02
1.02
Ka∗ /a0
1.015
1.01
1.005
1
0
0.1
0.2
µ
0.3
0.4
0.5
Figure 4: The optimal limit sell price for a risk-averse trader as a function of the price drift in presence of a random
delay in limit order execution. Results are shown for a0 = 1000, b0 = 995, ∆ = 1, T = 5 days, σ = 20%, ρ = 0.1,
λ = 8.
liquid market, there is no bid-ask spread (at = bt ) and the correlation is always unity. In a market with a random delay
higher correlation coefficient reduces gradually the optimal limit sell price the agent should submit (Fig. 7). If ask
price goes up then a limit order is more likely to get filled other things being equal. But if ask goes down and also
bid goes down then will have to sell at a lower price. therefore it makes sense that high ρ implies more passive order
submissions. This suggests a certain borderline value of ρ when one effect outweighs the other. When the drifts are
equal and there is a perfect correlation in bid and ask trajectories, we essentially deal with the situation of a constant
spread. It follows that the plot for a constant bid-ask spread case corresponds to that of no spread in Iori et al. (2003)
due to the fact that it becomes irrelevant.
Lastly, the order aggressiveness decreases as the maturity extends (Fig. 8). This result confirms previous statements that longer time horizons imply higher chances that at reaches the barrier represented by the trader’s limit price
Ka∗ . Again, this proves the consistency of our theoretical framework. Moreover, the optimal strategy of a risk averse
trader in an illiquid market is apparently less respondent to a change in expiry time.
6. Special Cases
We focus on two important special cases that allow us to separate two types of risks inherent to limit orders. First,
we examine the situation when the time constraint is removed. Second, we look into the limiting behavior of the
quotes chosen by the trader when the volatility of a traded asset is extremely low.
6.1. Infinite Time Horizon
Consider the case of infinitely long time horizon. We expect to detect a monotonic increasing relationship between
the limit price Ka and the expected return. The intuition behind this reasoning is that once the time pressure is removed,
the penalty which incurs paying the spread at maturity, simply vanishes. More precisely, we obtain
lim EGλ (γ) =
T →∞
λ
Ka
Kaγ
λ + γδ
a0
11
!A1 −A2 (γ)
.
(8)
1.025
ϕ=0
ϕ = 0.01
ϕ = 0.02
1.02
Ka∗ /a0
1.015
1.01
1.005
1
0
10
20
30
s0
40
Figure 5: The optimal limit sell price for a risk-averse trader as a function of initial spread s0 (in ticks). Results are
shown for a0 = 1000, b0 = a0 − s0 , ∆ = 1, T = 5 days, µ = 10%, σ = 20%, ρ = 0.1.
1.02
ϕ=0
ϕ = 0.01
ϕ = 0.02
Ka∗ /a0
1.015
1.01
1.005
1
−5
0
5
10
λ
Figure 6: The optimal limit sell price for a risk-averse trader as a function of market liquidity (log2 -scale). Results
are shown for a0 = 1000, b0 = 995, ∆ = 1, T = 5 days, µ = 10%, σ = 20%, ρ = 0.1.
12
1.02
ϕ=0
ϕ = 0.01
ϕ = 0.02
Ka∗ /a0
1.015
1.01
1.005
1
0
0.2
0.4
ρ
0.6
0.8
1
Figure 7: The optimal limit sell price for a risk-averse trader as a function of the bid and ask correlation in presence
of a random delay in limit order execution. Results are shown for a0 = 1000, b0 = 995, ∆ = 1, T = 5 days,
µ = 10%, σ = 20%, λ = 8.
1.03
ϕ=0
ϕ = 0.01
1.025
ϕ = 0.02
Ka∗ /a0
1.02
1.015
1.01
1.005
1
0
2
4
T
6
8
10
The optimal limit sell price for a risk-averse trader as a function of the expiry time in presence of a
random delay in limit order execution. Results are shown for a0 = 1000, b0 = 995, ∆ = 1, µ = 10%, σ = 20%,
ρ = 0.1, λ = 8.
Figure 8:
13
6.2. No Price Uncertainty
When the volatility of a security which the trader has to liquidate approaches zero, the price of this security will
be changing at a constant rate per unit of time. Therefore, the maximum that the best ask can attain until maturity is
known to be aT = a0 eµa T , and the best bid is bT = b0 eµb T .
lim EGλ (γ) =
σ→0
λ
Kaγ e−γδτ − eλτ · e−(λ+γδ)T + bγ0 e(γµb −γδ−λ)T · eλτ .
λ + γδ
(9)
a /a0 )
The moment when the limit sell order Ka will hit the quote is calculated as τ = ln(K
µa −δ . In this situation the trader
bears no price risk and the only risk he faces is linked to the non-execution of his orders due to delays. Considering
the deterministic nature of prices, the trader will optimize his strategy only over the subset of limit prices from the
interval a0 ≤ Ka ≤ a0 · e(µa −δ)T to ensure that τ ≤ T .
7. The Distribution of Waiting Times
The probability density function of a time-to-fill, as depicted in Fig. 9b, follows an exponential distribution with
a peak. The relationship between the distribution of a trading time in a market without delay and in a market with a
random delay is stated in the following proposition.
(b)
(a)
1.5
1
λ=∞
λ = 48
λ=8
λ=1
0.8
1
P (θ ≤ t)
P (θ ∈ dt)
0.6
0.4
0.2
0
0
0.5
λ=∞
λ = 48
λ=8
λ=1
0
2
4
t
6
8
10
0
2
4
t
6
8
10
Figure 9: The distribution of the time-to-fill of a limit order: (a) the cumulative distribution, (b) the probability
density function. Results are shown for a0 = 1000, b0 = 995, Ka /a0 = 1.01, µ = 10%, σ = 20%.
14
(b)
(a)
10
140
120
100
T
λ=∞
λ=8
λ=1
8
T
λ=∞
λ=8
λ=1
6
θ̄
θ̂
80
60
4
40
2
20
0
1
1.005
1.01
1.015
1.02
0
1
1.025
Ka /a0
1.005
1.01
1.015
1.02
1.025
Ka /a0
The characteristics of the most common time-to-fill of sell limit orders submitted at ifferent prices
Ka ≥ a0 : (a) the average time-to-fill, (b) the median time-to-fill. Results are shown for a0 = 1000, b0 = 995,
µ = 10%, σ = 20%.
Figure 10:
Proposition 3. Let P (τ ≤ t) be the cumulative distribution of limit order time-to-fill in a perfectly liquid market, then
the cumulative distribution function of waiting times in a market with small average delay P (θ ≤ t) is approximately
!
x − A1 σ2a t
x
P (τ ≤ t) − P (θ ≤ t) = ε̄
n
.
(10)
√
√
σa t t
σa t
The relationship between the probability density functions for two types of markets is:


x2 − A21 σ4a t2 − 3t 


 .
P (θ ∈ dt) = P (τ ∈ dt) 1 − ε̄
2σ2a t2
(11)
Proof. See Appendix A.3.
Since random delay ε follows exponential distribution, this proposition reveals that the discrepancy between waiting time in a market with infinite liquidity and a less liquid market is subject to an exponential component. The
result complies with the intuition that the difference in limit order execution is most sensitive to market liquidity when
medium-length maturities are concerned; both equations (10) and (11) show a negative dependence on a time factor.
In addition to that, as Figure 10a reveals, the average waiting time θ̄ grows linearly with the distance-to-fill of a
limit order. However, the contribution of the market liquidity parameter λ is negligible. Given that the distribution of
the time-to-fill is considerably skewed, the mean is not an informative statistic. We plot the median of the time-to-fill
θ̂ – a-50%-probability outcome. For instance, if a trader operating in a perfectly liquid market submits a sell order at
Ka = 1020, then it is equally likely to execute earlier or later than his horizon T = 5 days; whereas, in a market with
15
imperfect liquidity λ = 1 he must use a more aggressive order Ka = 1017 to achieve this. As Fig. 10b demonstrates,
the median trading time increases as passive orders fill the book. Also, the higher is the liquidity (lower λ), the shorter
is the median time-to-fill.
8. The Equilibrium Spread
It follows from the preceding discussion that in a continuous double auction the optimal decision of every market
participant is made on the basis of existing best bid and ask prices which define the market spread. Meanwhile, the
choices and actions that traders take now affect the current market spread and, as a direct consequence, influence
future trading decisions. In fact, the market spread is the result of the interactions between heterogeneous agents who
populate the market.
In the quote-driven markets the bid-ask spread is charged by the market maker in order to cover the expenses
incurred by trading against better informed agents. The reasons for the bid-ask spread in a double-auction markets
are more subtle; its presence is justified in several ways in the literature: information asymmetry which we have
discussed before, gravitational pull effect depicted by Cohen et al. (1981), variations in the state of the book and
traders valuations for the asset, trading costs, which in turn comprise direct costs such as commissions, transfer taxes
order submission fees and account service fees, and indirect cost – the difference between the price at which the
transaction was actually carried and a certain fair price. The delay parameter absorbs all these nuances which, in
essence, make the difference between trading with a limit order rather than a market order.
Cohen et al. (1981) examine this issue in detail and show that a positive market spread is incumbent to this market
microstructure. According to Cohen et al. (1981), the equilibrium spread in a dynamic trading system is “the bid-ask
spread at which, for the next instant of time, the probability of the spread increasing is equal to the probability of the
spread decreasing.” Adopting his definition to our framework we arrive at the condition (see Appendix B.1.1):
µa bt
= .
µb at
(12)
This condition, as Cohen et al. (1981) emphasize, does not guarantee that the market will eventually settle at this
equality, rather that it is more likely that the price will move towards this condition than in the opposite direction.
Since the drift parameters are constant, the condition in equation (12) simply states the equilibrium level of spread is
constant and is determined by the gap between the growth rates of bid and ask prices. However, in order to preserve
non-negative spread we must have at ≥ bt , ∀t, or µa ≤ µb in equilibrium. This slightly counter-intuitive contingency
arises from the specific behavior of log-normal process – an increase and a decrease by the same amounts are not
equally likely. If we impose for the purpose of stationarity µa = µb then we must observe a zero absolute spread in
equilibrium – equation (12) requires at = bt , ∀t.
Nevertheless, this definition does not account for the notion of delay which is the crucial characteristic of the
market we examine. It is precisely the deviation from the perfect liquidity case inflicted by large delays in limit order
execution that determines the real spread size.
Definition (Equilibrium Spread). In a dynamic trading process the equilibrium market spread is the bid-ask spread
such that the expected utility from trading via a limit order at the optimal limit price is equal to the utility from an
immediate market order.11
Applying numerical optimization we find the implied equilibrium spread given the expected delay.12 It has been
discussed in the empirical literature that the main determinants of the spread size are competition for liquidity and risk
aversion of market participants (Ranaldo, 2004). The higher is the competition among traders to provide liquidity,
the tighter is the observed bid-ask spread, whereas the degree of risk aversion of the traders has a positive impact on
the spread size. As depicted in Fig. 11a, the size of spread diminishes and eventually attains zero as the value of λ
11 This is an interpretation of a definition suggested by Harris (2003) (p.304): “The spread which ensures that traders are indifferent between
using a limit order and a market order is the equilibrium spread.”
12 The necessary derivations are presented in Appendix B.1.2.
16
Figure 11: Equilibrium spread a s a function of liquidity and risk tolerance of agents: (a) for different value of delay
intensity λ, (b) for different levels of risk aversion ϕ. Results are shown for b0 = 1000, a0 = 1005, Ka /a0 = 1.01,
µ = 10%, σ = 20%, ρ = 0, λ = 8.
increases, thereby decreasing the expected delay. In Fig. 11b we can see that the equilibrium spread increases, as
expected, with the degree of risk aversion ϕ.
Although this result is obtained from the standpoint of a seller, the pattern for the buyer would be symmetrical in
our setting. Moreover, since the limit orders are convertible in our model, seller has to monitor both sides of the book.
In other words, a market design where a limit order is executed immediately the moment it becomes the best price in
the market requires a zero bid-ask spread.
9. Conclusions
We have developed a logically consistent and empirically plausible model, which is easy to estimate. The central
feature of the present study is the analysis of the impact of a random delay in limit order execution on the optimal
strategy of a risk-averse trader. Based on an analytical solution, we examine the effects of various market parameters
on his optimal selling strategy.
Our framework both benefits from transparency and explains the trade-off between immediacy and a favorable
transaction price. In contrast with standard first passage time models of trading, it captures the fundamental difference between time required to reach the beginning of the queue on the relevant side of the market and the timeto-completion of a trade. The probability density of the expected time-to-fill of limit orders sharpens as liquidity
increases and reveals an empirically observed exponential distribution of trading times. The discrepancy, as the model
confirms, is due to imperfect liquidity, which in turn defers the trade execution. The main result suggests that the
introduction of a random delay factor alleviates the impact of various market conditions on the optimal limit price
the trader submits. Using comparative statics analysis, we demonstrate that the presence of a lag factor linearizes
these effects. Notably, it is not the magnitude but the mere presence of delay that alters the nature of the relationship.
Furthermore, we determine the equilibrium market spread as the bid-ask spread such that the expected utility from
trading via a limit order at the optimal limit price is equal to the profit from an immediate market order. We subsequently prove that, consistent with real market phenomena, the equilibrium bid-ask spread increases both as liquidity
decreases and agents’ risk aversion increases.
In addition to that we have demonstrated that in distinction from the profit-maximizing case, the introduction of
risk-aversion factor provides the results which are more coherent with empirical observations and seem to be more
useful for the practical implementation. The mean-variance utility function permits adequate risk assessment for a
strategy involving limit order trading, therefore our approach allows to model the trading trajectories of heterogeneous
investors.
We provide a solution for a static problem which can be extended to multi-period submission steps and solved
in a manner of Harris (1998). However, we expect that results will not alter qualitatively once a trader is allowed to
17
revise his strategy a finite number of times. Whereas we analyzed only small trades, large trades should be examined
differently since they have a price impact when market orders are used to execute the trade. There is a separate branch
of literature on the order splitting issues which are closely related to our framework (Almgren and Chriss, 2000;
Obizhaeva and Wang, 2005; Alfonsi et al., 2010; Løkka, 2011). In a recent paper by Guéant et al. (2011), the authors
propose a novel approach of splitting a large trade using limit orders rather than market orders. This is the direction
for the future development of our framework.
Appendix A. Proofs
Appendix A.1. Proof of Proposition 1
In order to
h find the analytical expression for ithe power of the limit order strategy payoff we need to calculate
EGλ (γ) ≡ E Kaγ e−γδ(τ+ε) I{τ+ε≤T } + bγT e−γδT I{τ+ε>T } .
We write a log-normal process as at = a0 eXt with Xt = µa − σ2a /2 t + σa Wt . Then the first passage time is
τ = inf{ t ≥ 0| Xt = x} with x = ln (Ka /a0 ). Denote the maximum of the Xt over a time period T as MTX = max{ Xt | 0 ≤
t ≤ T }. By the reflection principle ∀ x ≥ 0 we have
P (τ ≤ t) = P MtX ≥ x .
The cumulative distribution of the maximum of a Brownian motion with drift ξ and volatility ν and x ≥ 0 is equal to
!
!
−x − ξt
x − ξt
X
2ξx/ν2
,
(A.1)
N
P Mt ≤ x = N
√ −e
√
ν t
ν t
and
P (τ ∈ dt) = √
|x|
2πν2 t3
e
−(x−ξt)2
2ν2 t
.
(A.2)
i
h
i
h
We want to to calculate J1 ≡ E Kaγ e−γδ(τ+ε) I{τ+ε≤T } and J2 ≡ E bγT e−γδT I{τ+ε>T } . Assuming that the sopping
time and delays are independent and implementing the integrated expectations formula13 we arrive at the following
expression:
h
i
h
h
ii
J1 = Kaγ E e−γδ(τ+ε) I{τ≤T } I{ε≤T −τ} = Kaγ E e−γδτ I{τ≤T } E e−γδε I{ε≤T −τ} τ
Z T
Z T −τ
= Kaγ
e−γδτ f (τ)
e−γδε f (ε) dεdτ.
(A.3)
0
0
Since in our framework the prices follow geometric Brownian motion, the first passage time has an inverse Gaussian distribution with the probability density function f (τ) given in (A.2); the delay variable follows an exponential
distribution with a positive parameter λ, then f (ε) = λe−λε . Therefore, we simplify the integral in (A.4):
"Z T −τ
#
Z T
Z T −τ
Z T
e−γδτ f (τ)
e−γδε f (ε) dεdτ =
e−γδτ f (τ)
e−γδε λe−λε dε dτ
0
0
0
=
=
13 E(X)
0
λ e−γδτ f (τ)
1 − e−(λ+γδ)(T −τ) dτ
λ + γδ
0
"Z T
#
Z T
λ
e−γδτ f (τ) dτ − e−(λ+γδ)T
eλτ f (τ) dτ .
λ + γδ 0
0
Z
T
= E(E( X| Y))
18
Using equation (A.2) with ν = σa and ξ = µa − σ2a /2 we rewrite the first term as

 
Z


Z T
 T
 x − µa − σ2a /2 τ 
 x − µa − σ2a /2 τ  
λ
|x|
|x|

 dτ − e−(λ+γδ)T
 dτ .
e−γδτ
eλτ
J1 = Kaγ
√ n 
√ n 
√
√

 
λ + γδ  0
σa τ τ 
σa τ
σa τ τ 
σa τ
0
(A.4)
Since the following equality holds for the normal density


p
!
√
 α + t β2 + 2φ 
α + βt
−φt
−αβ+α β2 +2φ 
e n √
(A.5)
=e
n 
 ,
√
t
t
with α = x/σa , β = −(µa − σ2a /2)/σa and φ = γδ we rearrange the first component of J1 as


"
!#
Z T
Z T
 x − µa − σ2a /2 τ 
x + A2 σ2a τ
1
|x|
x(A1 +A2 )
 dτ = |x|
e−γδτ
e
n
dτ,
√ n 
√
√
√

σa 0 τ τ
σa τ τ
σa τ
σa τ
0
√
µ −σ2 /2
(µa −σ2a /2)2 +2γδσ2a
.
σ2a
where A1 = a σ2a and A2 =
a
Finally, using the identity
!
"
!
!#
√
√
α + βt
1
− |α|
− |α|
1
−2αβ
dt =
N √ − sgn (α) β T + e
N √ + sgn (α) β T ,
√ n √
|α|
t t
t
T
T
T
Z
0
(A.6)
substituting α = x/σa , β = A2 σa and Ka ≥ a0 , we get
|x|
σa
Z
T
"
1
√
τ τ
0
e x(A1 +A2 ) n
x + A2 σ2a τ
√
σa τ
!#
dτ =
Ka
a0
!A1 +A2
N
!
!A −A
!
−x − A2 σ2a T
−x + A2 σ2a T
Ka 1 2
+
N
.
√
√
a0
σa T
σa T
The second integral in J1 is modified via the following identity which holds for the values λ ≤ β2 /2:14


p
!
√
 α + t β2 − 2λ 
α + βt
λt
−αβ+α β2 −2λ 

 .
e n √
=e
n 
√
t
t
(A.7)
(A.8)
Precisely, we obtain
T
Z
0
√
where A3 =
|x|
σa
Z
T
14 eλt n α+βt
√
t
(µa −σ2a /2)2 −2λσ2a
.
σ2a
1
√
τ τ
0
=


"
!#
Z T
 x − µa − σ2a /2 τ 
x + A3 σ2a τ
1
|x|
x(A1 +A3 )
 dτ = |x|
e
n
dτ,
eλτ
√ n 
√
√
√

σa 0 τ τ
σa τ τ
σa τ
σa τ
"
e
x(A1 +A3 )
x + A3 σ2a τ
n
√
σa τ
2
1 α+βt
√
t
−
eλt √1 e 2
2π
Applying formula (A.6) with α = x/σa , β = A3 σa and Ka ≥ a0
=
√1 e−
2π
!#
Ka
dτ =
a0
α2 +β2 t2 +2αβt−2λt2
2t
=
!A1 +A3
!
!A −A
!
−x − A3 σ2a T
−x + A3 σ2a T
Ka 1 3
N
+
N
.
√
√
a0
σa T
σa T
 √
2
√
 α+ β2 −2λt 
 −αβ+α β2 −2λt
√
− 12 
t
1
√ e
2π
19
√
= e−αβ+α
√
2
β2 −2λ n α+t √β −2λ
t
!
(A.9)
Combining the results (A.7) and (A.9) we arrive that at the final expression for the first term:

!A +A
!
!A −A
!
−x − A2 σ2a T
−x + A2 σ2a T
λ  Ka 1 2
Ka 1 2

N
+
N
√
√
λ + γδ a0
a0
σa T
σa T

!
!
!A1 +A3
!A1 −A3
2
 Ka
−x − A3 σa T
−x + A3 σ2a T 
Ka
−(λ+γδ)T 
 .
+
− e
N
N
√
√

a0
a0
σa T
σa T
Kaγ
=
J1
Before calculating the second term we rewrite the bid and ask equations to express these processes in terms of a
two-dimensional Brownian motion (W1 , W2 ):
dat = at µa dt + σ̄a dW̄t
dbt = bt µb dt + σ̄b dW̄t ,
p
where σ̄a = (σa , 0) and σ̄b = σb ρ, σb 1 − ρ2 . Therefore,
2
at = a0 e(µa −|σ¯a | /2)t+σ¯a W̄t
2
bt = b0 e(µb −|σ¯b | /2)t+σ¯b W̄t .
Notice that if bt is a log-normal process then bγt is also log-normally distributed. Applying Ito’s formula to this
process we can show that
!
γ(γ − 1) 2
γ
γ
σb dt + γbγt σ̄b dW̄t .
(A.10)
dbt = bt γµb +
2
Hence, applying Ito’s lemma once again we get bγT = bγ0 e(γµb +γ(γ−1)σb /2)T +γσ¯b W̄T . Now we rewrite the second term
2
J2
=
=
=
=
γ(γ−1) 2
e−γδT E bγ0 e γµb + 2 σb T +γσ¯b W̄T · I{ε>T −τ} · I{τ>T } + I{τ≤T }
2
bγ0 eγ(µb +(γ−1)σb /2−δ)T E ∗ I{τ>T } + I{ε>T −τ} I{τ≤T }
h
h
ii
2
bγ0 eγ(µb +(γ−1)σb /2−δ)T E ∗ I{τ>T } + E ∗ I{τ≤T } E ∗ I{ε>T −τ} τ
!
Z ∞
Z T
Z ∞
γ γ(µb +(γ−1)σ2b /2−δ)T
b0 e
f (τ) dτ +
f (τ)
f (ε) dεdτ ,
0
T
(A.11)
T −τ
∗
where expectation is calculated under probability measure P∗ defined by Radon-Nikodym derivative ηT = dP
dP =
σ̄b WT −|σ¯b |2 T
∗
∗
e
. As Girsanov theorem states, W̄t = W̄t − γσ̄a t is a two-dimensional Brownian motion under P , so we
rewrite
XT = (µa − σ2a /2)T + σ̄a W̄T∗ − γσ̄a T = (µa + (γ − 1/2)σ2a )T + σ̄a W̄T∗ ,
implying a drift ξ∗ = µa − σ2a /2 + γρσa σb under P∗ .
The first integral in (A.11) is the probability that the first passage time exceeds horizon T , which is given in
formula (A.1) for ν = σa and ξ∗ :


!
!2A
!
Z ∞
 x − µa − σ2a /2 + γρσa σb τ 
x − A4 σ2a T
−x − A4 σ2a T
Ka 4
|x|




−
N
,
(A.12)
√ n 
√
√
√
 dτ = N
a0
σa τ
T σa τ τ
σa T
σa T
where A4 =
Z
µa −σ2a /2+γρσa σb
.
σ2a
T
Z
∞
f (τ)
0
Furthermore, simplifying the last term in (A.11) yields
T −τ
f (ε) dεdτ =
Z
T
"Z
∞
f (τ)
0
#
Z
λe−λε dε dτ =
T −τ
0
20
T
f (τ) e−λ(T −τ) dτ = e−λT
Z
0
T
eλτ f (τ) dτ.
We substitute the probability density function f (τ) with volatility σa and drift ξ∗ = µa − σ2a /2 + γρσa σb and using
property (A.8) obtain


"
!#
Z
Z T
 x − µa − σ2a /2 + γρσa σb τ 
x + A5 σ2a τ
|x|
|x| T 1
λτ
x(A
+A
)


4
5
 dτ =
e
n
dτ,
(A.13)
√ n 
√ e
√
√
σa 0 τ τ
σa τ τ 
σa τ
σa τ
0
√
(µa −σ2a /2+γρσa σb )2 −2λσ2a
and λ ≤ (µa − σ2a /2 + γρσa σb )2 /2σ2a . Given the parameters values α = x/σa ,
where A5 =
σ2a
β = A5 σa and Ka ≥ a0 we write an explicit expression for this integral as shown in formula (A.6)
|x|
σa
Z
0
T
"
!#
!A +A
!
!A −A
!
x + A5 σ2a τ
−x − A5 σ2a T
−x + A5 σ2a T
Ka 4 5
Ka 4 5
x(A4 +A5 )
e
n
dτ, =
+
.
N
N
√
√
√
a0
a0
τ τ
σa τ
σa T
σa T
1
√
(A.14)
Using (A.12) and (A.14),
J2
=
bγ0
+


e−λT 

!
!
!2A
 x − A4 σ2a T
−x − A4 σ2a T
Ka 4
N
−
N
√
√
a0
σa T
σa T
!
!A4 +A5
!A4 −A5
2 !
−x − A5 σa T
−x + A5 σ2a T 
Ka
Ka
 .
+
N
N
√
√
a0
a0
σa T
σa T
γ(µb +(γ−1)σ2b /2−δ)T
e
Thus,

!A +A
!
!A −A
!
−x − A2 σ2a T
−x + A2 σ2a T
λ  Ka 1 2
Ka 1 2

+
N
N
√
√
λ + γδ a0
a0
σa T
σa T

!
!
!
!A −A
 Ka A1 +A3
−x − A3 σ2a T
−x + A3 σ2a T 
Ka 1 3
−(λ+γδ)T 

+
− e
N
N
√
√

a0
a0
σa T
σa T

!
!2A
!
 x − A4 σ2a T
−x − A4 σ2a T
Ka 4
γ γ(µb +(γ−1)σ2b /2−δ)T 

+ b0 e
−
N
N
√
√
a0
σa T
σa T

!
!A −A
!
!
 Ka A4 +A5
−x − A5 σ2a T
−x + A5 σ2a T 
Ka 4 5
−λT 
N
N
+ e
+
√
√

 ,
a0
a0
σa T
σa T
√
√
√
(µa −σa2 /2)2 +2γδσa2
(µa −σa2 /2)2 −2λσa2
(µa −σa2 /2+γρσa σb )2 −2λσ2a
µa −σ2a /2
µa −σ2a /2+γρσa σb
with A1 = σ2 , A2 =
, A3 =
, A4 =
, A5 =
σ2a
σ2a
σ2a
σ2a
a
h i
(µa −σ2a /2)2 (µa −σ2a /2+γρσa σb )2
and ∀ λ ∈ 0, λ̄ , where λ̄ = min
,
.
2σ2
2σ2
EGλ (γ) =
Kaγ
a
a
Appendix A.2. Proof of Proposition 2.
We find the limit of the expected utility using Laplace’s integral approximation method. Define an integral
I (λ) =
Z
b
h(τ)eλg(τ) dτ,
(A.15)
a
where [a, b] is a finite interval and functions h(τ) and g(τ) are continuous.
Lemma 1. Suppose the function g(τ) attains a maximum on [a, b] at either endpoint, τ0 = a or τ0 = b, and is
differentiable in a neighborhood of τ0 , with g0 (τ0 ) , 0 and h(τ0 ) , 0. Then the leading term of the asymptotic
expansion of the integral (A.15) , as λ → +∞, is given by
I (λ) =
h(τ0 ) · eλg(τ0 )
.
λ |g0 (τ0 )|
21
(A.16)
We apply this lemma to functions g(τ) = τ and h(τ) =
α+βτ
|α|
√ n
√
τ τ
τ
with integration limits a = 0, b = T . The
function g(τ) attains the highest value at τ0 = b = T and g (τ0 ) = 1, therefore we obtain from formula (A.16):
α+βT
|α|
!
Z T
√ n
√
· eλT
α + βτ
T T
T
|α|
eλτ √ n √
dτ =
λ · |1|
τ τ
τ
0
!
α + βT
|α|
·
n
= eλT
(A.17)
√
√
λT T
T
0
We use this result to approximate the second integral in (A.4) and the integral in (A.13) when λ ≥ β2 /2 and
identity (A.8) does not hold. Let ε̄ = E [ε] = 1/λ, then it is easy to show that for high values of λ the power utility is
approximately

!
!
!A −A
!
 Ka A1 +A2
−x − A2 σ2a T
−x + A2 σ2a T
1
Ka 1 2

EGλ (γ) N
+
N
√
√
1 + γδε̄ a0
a0
σa T
σa T
2 !#
x − A1 σa T
x
− ε̄ e−γδT
(A.18)
√ n
√
σa T T
σ T
 a
!2A
!
!
 x − A4 σ2a T
−x − A4 σ2a T
Ka 4
2
N
−
+ bγ0 eγ(µb +(γ−1)σb /2−δ)T N
√
√
a0
σa T
σa T
2 !#
x − A4 σa T
x
,
+ ε̄
√ n
√
σa T T
σa T
√
K (µa −σ2a /2)2 +2γδσ2a
µ −σ2 /2+γρσ σ
µa −σ2a /2
a
where x = ln a0 , A1 = σ2 , A2 =
, A4 = a a σ2 a b . Further, noting that constants A1 , A2 and
σ2a
a
a
A4 are independent of ε̄, in a perfectly liquid market we obtain
Kaγ
lim+ EGλ (γ) =
ε̄→0
+

!
!
!
!A −A
 Ka A1 +A2
−x − A2 σ2a T
−x + A2 σ2a T 
Ka 1 2

+
N
N
Kaγ 
√
√
a0
a0
σa T
σa T

!
!2A4
!
 x − A4 σ2a T
−x − A4 σ2a T 
K
a
γ γ(µb +(γ−1)σ2b /2−δ)T 
N
 .
b0 e
−
N
√
√
a0
σa T
σa T
Appendix A.3. Proof of Proposition 3
In the presence of a random delay in limit order execution we denote time-to-fill as θ = τ + ε and obtain its
distribution:
Z t
Z t−τ
P (θ ≤ t) =
f (τ)
f (ε)dεdτ
0
0
Z t
Z t−τ
Z t
=
f (τ)
λe−λε dεdτ =
f (τ) 1 − e−λ(t−τ) dτ
0
0
0
Z t
Z t
=
f (τ)dτ − e−λt
eλτ f (τ)dτ
0
0
!
x − A1 σ2a t
x
,
(A.19)
= P (τ ≤ t) − ε̄
√ n
√
σa t t
σa t
µ −σ2 /2
under on the probability space (P, Ω, F ) with x = ln Ka0a , A1 = a σ2a and ε̄ = λ1 .
a
Note that the probability density of first passage time P(τ ≤ t) is defined in equation (A.2). Taking the derivative
of the second term in expression (A.19) with respect to t, we arrive at the probability density function of limit order
22
time-to-fill θ



!
x2 − A21 σ4a t2 − 3t 
x2 − A21 σ4a t2 − 3t 

x − A1 σ2a t 


 .

P (θ ∈ dt) =
√ n
√
 = P (τ ∈ dt) 1 − ε̄
1 − ε̄
2σ2a t2
2σ2a t2
σa t t
σa t
x
(A.20)
It follows immediately from (A.20) that when the expected delay approaches zero time-to-fill is equivalent to the time
it takes to reach the front of the queue: limε̄→0+ P (θ ∈ dt) = P (τ ∈ dt).
Taking into account that prices are independent of delays, we have E [θ] = E [τ + ε] = E [τ] + E [ε]. The mean of
an inverse Gaussian given in (A.2) is x/ξ, the mean value of exponential delays is 1/λ, therefore we find the average
waiting time-to-fill of a limit order:
ln Ka0a
1
+ .
(A.21)
E [θ] =
2
µa − σa /2 λ
The mode of he sum of two independent variable is a sum of their modes and since the mode of exponentially
distributed random variable is zero, the mode of the time-to-fill is:
1/2
2 2
4 ln Ka0a µa − σ2a /2 + 9σ4a
− 3σ2a
.
(A.22)
Mode [θ] =
2 µa − σ2a /2 2
Appendix B. Additional Derivations
Appendix B.1. The Equilibrium Spread Derivations
Appendix B.1.1. Derivation of Condition (12).
Denote S t ≡ at − bt , then the probability that the spread will increase in an infinitely small time increment dt is
equivalent to saying that the change in ask will be greater than the change in bid, independently of the direction.
h
i
Pr [dat ≥ dbt ] = Pr at µa dt + σ̄a dWta ≥ bt µb dt + σ̄b dWtb
!
!
#
"
q
√
√
bt
2
µb dt + ρσb ψ1 + σb 1 − ρ ψ2 dt ≤ µa dt + σa ψ1 dt
= Pr
at
"
!
!
#
q
√
bt
bt
bt
2
= Pr
ρσb − σa ψ1 + σb 1 − ρ ψ2 ≤ µa − µb dt
at
at
at
= Pr ψ3 ≤ zt = N (zt ) ,
where ψ j ∼ N (0, 1) are independent random variables, zt ≡
µa −µb bt /at
νs
√
dt with ν2s =
b2t 2
σ
a2t b
+ σ2a − 2ρσa σb abtt . It is easy
to see that variance ν2s is strictly positive for non-zero volatilities of the asset prices σa and σb and |ρ| ≤ 1.
Similarly, the probability of a narrower spread is expressed as
Pr [dat ≤ dbt ] = Pr ψ3 ≥ zt = Pr ψ3 ≤ −zt = N (−zt ) = 1 − N (zt ) .
Equating two expressions yields:
Pr [dat ≥ dbt ] = Pr [dat ≥ dbt ]
N (zt ) = 1 − N (zt )
bt
µa
=
.
µb
at
Appendix B.1.2. Numerical Calculation of the Spread in a Market with Delay.
The preferences of a trader are described by a mean-variance utility function (5). The equilibrium spread S̃ is
defined as a situation when the expected utility from using a limit order at the optimal price Ka∗ equals the utility of
23
the profit from trading via immediate market order. Let S̃ = at − bt , ∀, then
S̃ = a0 − EU Ka∗ ; S̃ .
(B.1)
Using the approximation from the previous section (A.19) we substitute b0 = a0 − S and find an optimal limit price as
a function of spread Ka∗ = Ka∗ (S) for a range S ≥ 0. Then we determine the pair of limit price and spread that satisfy
condition (B.1).
Appendix B.2. Bounds of the Expected Payoff Power Utility
The general form of the integral we need to bound is
Z
T
e−λ(T −τ) f (τ)dτ,
(B.2)
0
where the density is f (τ) =
α+βτ
|α|
√ n
√
τ τ
τ
and λ takes large values. It is easy to see that
e−λ(T +τ) ≤ e−λ(T −τ) ≤ 1,
∀τ ∈ [0, T ] :
and since f (τ) is a bounded function we have
Z T
Z
e−λ(T +τ) f (τ)dτ ≤
0
T
Z
e−λ(T −τ) f (τ)dτ ≤
0
T
f (τ)dτ.
0
For the upper bound we apply identity (A.6)
Z
T
f (τ)dτ
=
0
=
!
α + βτ
|α|
dτ
√ n √
τ
0 τ τ
!
!
−α + βT
−α − βT
N
+ e−2αβ N
.
√
√
T
T
Z
T
(B.3)
In order to calculate the lower bound we first use property (A.5)



p
"
!#
√2
 α + τ β2 + 2λ 
α + βτ
|α| 
|α|
 ,
= e−λT √ e−αβ+α β +2λ n 
e−λ(T +τ) f (τ) = e−λT √ e−λτ n √
√
τ τ
τ
τ τ
τ
then property (A.6) to get
Z
T
−λ(T +τ)
e
f (τ)dτ
=
0
=
where ψ =
!
α + ψτ
1
dτ
|α| e
√
√ n
τ
0 τ τ
!
!#
"
−α + ψT
−α − ψT
−α(ψ+β)
−λT
α(ψ−β)
+e
N
,
e
e
N
√
√
T
T
α(ψ−β)−λT
Z
p
β2 + 2λ.
24
T
(B.4)
The full expression for expected power utility is
EGλ (γ) =
−
+
+

Z

 T
 x − µa − σ2a /2 τ 
λ
|x|

 dτ
e−γδτ
Kaγ
√ n 
√

λ + γδ  0
σa τ τ 
σa τ



Z T
 x − µa − σ2a /2 τ  
|x|
λτ
−(λ+γδ)T
 dτ
e
e
√ n 
√
 
σa τ τ 
σa τ
0

Z

 ∞ |x|
 x − µa − σ2a /2 + γρσa σb τ 
γ γ(µb +(γ−1)σ2b /2−δ)T 

 dτ
b0 e
√ n 
√


σa τ
T σa τ τ
 

Z T
2
 x − µa − σa /2 + γρσa σb τ  
|x|
λτ
−λT
 dτ .
e
e
√ n 
√
σa τ τ 
σa τ
0
As shown in the the proof of Proposition 1, the first and the third integrals have analytical expression for all values
of parameters. Since the second term in this expression is negative with α = x/σa and β = −A1 σa we apply formula
(B.4) to determine its upper bound

!
!
!A −A
!
 Ka A1 +A6
−x + A6 σ2a T
−x − A6 σ2a T 
Ka 1 6
 ,
I2 ≤ −e−(λ+γδ)T 
N
+
N
√
√
a0
a0
σa T
σa T
√
where A6 =
(µa −σ2a /2)2 +2λσ2a
,
σ2a
(B.5)
and formula (B.3) to determine its lower bound
−γδT
I2 ≥ −e

!
!2A1
!
 −x − A1 σ2a T
−x + A1 σ2a T 
K
a
N
 .
+
N
√
√
a0
σa T
σa T
(B.6)
Similarly, for the fourth term in the expected power utility function α = x/σa and β = −A4 σa we apply formula
(B.3) to determine its upper bound
!2A
!
!
−x − A4 σ2a T
−x + A4 σ2a T
Ka 4
I4 ≤ N
N
+
√
√
a0
σa T
σa T
and (B.4) to find the lower bound

!
!
!A −A
!
 Ka A4 +A7
−x + A7 σ2a T
−x − A7 σ2a T 
Ka 4 7
 ,
N
+
N
I4 ≥ e−λT 
√
√
a0
a0
σa T
σa T
√
(µa −σ2a /2+γρσa σb )2 +2λσ2a
where A7 =
.
σ2a
Finally, substituting (B.5) and (B.7) we obtain an upper bound of expected power utility:
EGU
λ (γ)
=
−
+

!A +A
!
!A −A
!
−x − A2 σ2a T
−x + A2 σ2a T
λ  Ka 1 2
Ka 1 2

N
N
+
√
√
λ + γδ a0
a0
σa T
σa T

!
!
!A −A
!
 Ka A1 +A6
−x + A6 σ2a T
−x − A6 σ2a T 
Ka 1 6

e−(λ+γδ)T 
N
+
N
√
√
a0
a0
σa T
σa T
Kaγ
bγ0 eγ(µb +(γ−1)σb /2−δ)T .
2
25
(B.7)
(B.8)
Using results (B.6) and (B.8) we obtain a lower bound of expected power utility:

!A +A
!
!A −A
!
−x − A2 σ2a T
−x + A2 σ2a T
λ  Ka 1 2
Ka 1 2

N
+
N
√
√
λ + γδ a0
a0
σa T
σa T

!
!2A1
!
2
2
 −x − A1 σa T
−x + A1 σa T 
Ka

− e−γδT N
+
N
√
√
a0
σa T
σa T

!
!2A
!
 x − A4 σ2a T
−x − A4 σ2a T
Ka 4
γ γ(µb +(γ−1)σ2b /2−δ)T 

+ b0 e
−
N
N
√
√
a0
σa T
σa T

!
!
!
!A −A
 Ka A4 +A7
−x + A7 σ2a T
−x − A7 σ2a T 
Ka 4 7
−λT 
+ e
+
N
N
√
√


a0
a0
σa T
σa T
√
√
√
(µa −σa2 /2)2 +2γδσa2
(µa −σa2 /2)2 +2λσa2
(µa −σa2 /2+γρσa σb )2 +2λσ2a
µa −σ2a /2
µa −σ2a /2+γρσa σb
with A1 = σ2 , A2 =
, A4 =
, A6 =
, A7 =
.
σ2
σ2
σ2
σ2
EGλL (γ) =
a
Kaγ
a
a
a
a
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